Focus on Math

Helping children become mathematicians!

Twain Sullivan Elementary, Houston, BC April 27, 2013

Filed under: General Math — Focus on Math @ 4:35 pm

twain sullivan ele What a grand day we had at Twain Sullivan Elementary yesterday, in spite of a number of “complications” (including losing power in half the school and, oddly enough, half the room we were in). I did not even remember to tell the participants that when I first pulled into the school parking lot in the morning, there was a beautiful rainbow settled just off to the side of the school. It seemed a good omen for the day – there is something lovely about a rainbow!

We were able to persevere through the strange complications and ended up having great session. The main topic of the day was number sense, and we particularly focused on helping students build particular relationships between numbers, thus creating connections that would lead to greater overall math understanding.

There were a number of handouts that we used, and I promised “clean copies” of those. Some of those handouts are also posted elsewhere on the blog, but for the sake of simplicity, I shall repost them here in one spot for easy access for the Twain Sullivan folks (and anyone else who is interested):

Powerpoint handout of the day
Student ten frames
Mini blank ten frames 27 per page,  40 per page
Teacher ten frames
Large dot cards – set 1
Large dot cards – set 2
Large dot cards – set 3
100 dot arrays (4 & 12)
100 dot array (large)
mini blocks of 100 black dots
How far to 20?
How far to 30?
How far to 100?

The really large 10-frame that we used does not download properly – at least I have not figured out how to make it work.

If I have left anything off the list, let me know and I will add it on.

I hope I get to work with you all again in the future. I was impressed with your willingness to adapt what you are doing in math to make it better for your students. Thanks again for a great day in Houston, BC!

Mathematically yours,


An addition poster “pour mes amis francophone”. April 22, 2013

Filed under: General Math — Focus on Math @ 3:46 pm

French poster Merci, Maria L. for letting me share this poster you have made for your French immersion classroom. It is always good to have a visual reminder of strategies students can use for math thinking.

Mathematically yours,


NCTM 2013 Denver Presentation April 18, 2013

NCTM workshop picI am looking forward to a great hands-on session tomorrow as we are “Packing a Powerful Punch with Patterns” (presentation #500, located in the Hyatt Regency, Centennial Ballroom E beginning at 1:00). We will be focusing on how to help students make the transition from basic patterning skills to algebraic thinking, uncovering the deeper math that is embedded in patterning. Our vehicle will be growth patterns that we make out of pattern blocks. If you are here in Denver this week, I hope you are able to join us for the session.

The handouts given out in the session were a truncated version of the PowerPoint presentation, and as promised I am making the full version of the handout available here. If you use these in your classroom, I would love to hear from you about the lessons and even see some samples of student work, too.

Mathematically yours,

grow pat chart pic


Problem Solving in a Grade 1 Class April 16, 2013

sheep & chickens 2 I was in Mrs. Powers’ grade 1 class last week doing a problem-solving lesson during their math time. After reading the class the story How Many Feet in the Bed, we did a version of the question about the number of two-legged and four-legged animals possible for a given number of legs. The actual question read this: “In a pen on a farm were some chickens and some sheep. Devon was looking under the pen’s fence and counted 12 legs. How many animals could there be in the pen?” However, I realized when I got to Mrs. Powers’ class with the question ready to paste into student books that what I had wanted was for the question to indicate that the pen could hold chickens and/or sheep, thus allowing for four possible combinations and not just two. So Mrs. Powers and I talked to the children about the question and had them answer the intended question not the written one.

I should mention that we had a conversation before the class started to work on answering the questions about REPRESENTING our thinking. I often talk to children about their drawings when solving an answer – that we are doing MATH and not ART. The picture only needs to represent the things in the question in some way mot be a detailed drawing.

The students went to work with a will. Although ten frames and counters were also available, drawing was by far the method of choice used by almost all the children for solving the problem. Clearly the two students whose work you see here took me at my word about “representing” their thinking, and each drew “sketchy” chickens with two legs and “sketchy” sheep with four legs.

One student’s work shows all four solutions recorded correctly. The second sample also has four solutions, but two are the same. Looking closely, however, it looks to me like the solution in the upper left corner of that sample had the fourth correct answer of three sheep, but was erased.

Mrs. Powers was delighted with the children’s thinking. I predict the students will be doing more problem-solving in the weeks to come! Good job, class!

Mathematically yours,

sheep & chickens 1


Salt Spring Island Day 2 April 6, 2013

salt spring pic Day 2 was another busy day for me in the town of Ganges on the lovely Salt Spring Island. First off there was a three-hour workshop for Primary teachers. Our focus was on the big ideas about numbers that we should be working to establish in young children, and how those big ideas could be “layered” or built upon for older students. Those number relationships are a huge part of overall “number sense” which we hope students will build as they explore numbers in many different ways.

After lunch was another three-hour workshop, this time for Intermediate teachers. Delightfully, some of the participants from the first workshop were back for more math! We had a grand time, even if I must say so myself! Our focus was on strategies for doing the major operations (addition, subtraction, multiplication, and division) and tools that would support those operations.

Intersperced in both sessions were lots of examples, stories, and of course, the occasional “math woo” when we just could not contain our excitement. (Ok, I was the one doing the most woo-ing, but how can you not get excited over math!)

I mentioned to the participants that I would post some things for them, so here they are

large 100 dot array

small 100 dot arrays (in two sizes)

• mini blank 10 frames for students to use in problem solving: 27 per page40 per page

10 frames for students (with dots)

teacher 10 frames (with dots)

mini 100 blocks of ten frames

problem solving rubric

If I have missed something that I promised, please let me know and I will add it to the download list.

I had a wonderful time talking math with all those who came to the sessions both days. I hope I am privileged to come back for another round one day!

Mathematically yours,


Salt Spring Island: Day 1 April 4, 2013

salt spring air pic I boarded this float plane this morning for a quick flight from Vancouver to Salt Spring Island (flying Salt Spring Air) where I was greeted on the dock by my host for these two days, Kelda Logan. We hurried off to Gulf Islands Secondary School to get set up for the day’s session with middle and high school teachers.

My thanks to all the participants for their hard work as we made our way through the day’s content:
• Discussing what it means to “do math”;
• Looking at what we really want to see in our math students (regarding their knowledge and personal qualities)
• Looking at how to increase the use of mathematical processes in classrooms;
• Examining different tasks/activities/problems and looking at the “richness” (or lack thereof) they offer
• Discussing ways to build a culture in the classroom that supports deep thinking
• Using rubrics to evaluate problem solving, and to help students learn to self-evaluate their own work (download the rubric here)
• And, of course, we spent some time playing with numbers ourselves!

My thanks also to all of the parents who came to the parent meeting after school and who were game to play with numbers and look at math from a different perspective! Hopefully your brain, stretched today, will not be able to return to its original dimensions (to loosely quote Oliver Wendell Holmes)!

I am looking forward to two great sessions tomorrow, first with primary teachers, and then with intermediate teachers! So much math, so little time!!!

Mathematically yours,


The problem with our schools… April 1, 2013

Filed under: General Math,Ideas from Carollee's Workshops,Parents — Focus on Math @ 11:16 am

Screen shot 2013-03-29 at 4.09.23 PM The problem with our schools is not that they are not what they used to be, but that they are what they used to be. *

I saw this quote recently and wanted to write something about it as it struck a resonating chord in me. But before we talk about schools, let’s think about medicine for a just a moment. Who among us would go to a doctor who was still running his (or her) practice as if medical knowledge had not changed in recent decades? Do you not expect your physician to stay up-to-date on medical procedures rather than rely on what was common practice some time ago?

There is so much more knowledge now about the brain and learning than there ever has been before. (**If you are interested, you can read a bit about that at the end of the blog.) Why is it, then, that so many classrooms are stuck in practices and procedures that are the same as what was happening in classrooms decades ago? Why have we not expected teaching to change as science reveals more about how the brain learns? Rote memorization of bits of information (information that is, incidentally, available at the touch of our fingers these days) is not a wonderful learning experience. I think every teacher is exposed to Bloom’s taxonomy of learning in teacher training. Guess what the “lowest” level of learning is: knowledge, meaning the knowing of factual things. We need students who can think, can solve problems, can analyze situations and apply knowledge. The world is increasing in complexity, and we need to develop minds that are prepared for such a world.

Now, this is not just about technology, which, in and of itself is a rather neutral thing, in my opinion. Technology can be used in the classroom without there being changes in teaching practice. For instance, I have seen SmartBoards being used in classrooms as a kind of glorified worksheet, thus really not being used to change to instructional practices that are “brain-friendly”. On the other hand, there are many technologically outdated classrooms where students are doing deep thinking and learning.

When a teacher does step out and begin to change practice, often the first reaction from parents is to challenge the change. I see this time and time again as I work closely with teachers in the area of mathematics. Parents of students call up the teachers or send notes demanding to see pages of math work being sent home that look exactly like the pages being sent home when the parents were in school themselves. Parents wonder why things need to change. They wonder why there have to be so many words on the math page; why just getting a correct answer is not enough; why a student should justify his reasoning; why drawing a representation of a problem is important. In general, the parents want the math to look like it did when they were in school. They might even say, “That way was good enough for me, it is good enough for my kids!” Can you imagine someone saying that to a doctor, asking for an outdated procedure?!

Sometimes we need to really ask probing questions of ourselves, and I hope you will now do just that: Are you a teacher or parent who is “stuck in the past”, assuming that mathematics instruction should not change? Have you considered the science behind the changes? Have you asked how the changes can help your students or children?

Do you know some parents or teachers who would like to probe this idea further? Let’s arrange a math meeting!

Mathematically yours,

*I could not find a source to whom to give credit for the quote. If you know this, by all means, let me know and I will give due credit.

**Some notes on the science of learning.
In recent years we have learned so very much about the science of the brain and about how people learn. We have amazing tools such as Magnetic Resonance Imaging (MRI) machines, Functional MRI (f MRI) machines, and NMRI (Nuclear Magnetic Resonance Imagery) that help us “see” how people think and learn. The electroencephalogram (EEG) gives us readings about the electrical output of the brain. Magnetoencephalography (MEG) uses high-tech sensors that are super-cooled, liquid-helium, and superconductive to locate faint magnetic fields that are generated by the brain’s neural networks. They’ve been used to detect brainwave patterns in people. These tools also can help us track, for example, how much brain activity occurs during problem solving. Still further, there is Positron Emission Tomography (PET), an imaging device that reveals where certain areas of the brain are “working”. All of these tools have been combined with studies to reveal much about the brain learns. And these high-tech tools are just the beginning. The research about how people learn best is vast!